Combination Calculator -- nCr

In a combination, order does not matter: $\{A, B, C\}$ and $\{C, A, B\}$ count as the same selection. In a permutation, order matters: ABC and CAB are different arrangements. Combinations count subsets; permutations count sequences.

Use the formula $C(n, r) = \frac{n!}{r! \times (n - r)!}$. For example, $C(10, 3) = \frac{10!}{3! \times 7!} = \frac{720}{6} = 120$. This counts the number of ways to choose 3 items from 10 without regard to order.

“n choose r” (written as $C(n, r)$ or the binomial coefficient) counts how many ways you can select $r$ items from a set of $n$ distinct items when order does not matter. For example, “52 choose 5” gives 2,598,960 possible poker hands.

Use combinations with repetition when items can be chosen more than once, such as selecting scoops of ice cream from available flavors. The formula is $C(n + r - 1, r)$, also known as the stars and bars formula.

This symmetry property reflects the fact that choosing $r$ items to include is equivalent to choosing $n - r$ items to exclude. For example, picking 3 items from 10 ($C(10, 3) = 120$) is the same as picking 7 items to leave out ($C(10, 7) = 120$).